The spacing conjecture for the transformed zeta zeros

Let $\rho_k$ be the nontrivial zeros of the Riemann zeta function with positive imaginary part, and set

$$\eta_k=\operatorname{Im}(\rho_k)+i\left(\frac12-\operatorname{Re}(\rho_k)\right),\qquad \lambda_k=\eta_k^2.$$

Let $\Lambda$ be the associated entire function, and let its spacing constant be the infimum of the absolute differences of real parts of distinct nonconjugate zeros. Spacing conjecture. The spacing constant for $\Lambda$ is positive and equals $\lambda_6-\lambda_5$, with

$$\lambda_6-\lambda_5>159.$$

The conjecture is motivated by numerical evidence from the first two million zeros in the Odlyzko tables. Explicit spacing bounds are generally unavailable, and it remains open whether all zeta zeros are simple.